Scenario return and risk

• Expected returns \((E(R))\) are defined as: \[E(R) = \sum_{s=1}^K p_s \times R_s\]

  • \(s\) represents an economic scenario ranging from 1, 2, 3 … to \(K\)
    • The scenarios could be recession, boom or normal economic times
  • \(p_s\) is the probability of a certain scenario (\(s\)) occuring
  • \(R_s\) is the asset return if a certain scenario (\(s\)) occurs

• Variance is defined as: \[\sigma^2=\sum_{s=1}^K p_s \times [R_s-E(R)]^2\]

• Standard deviation \((\sigma)\) = \(\sqrt{\sigma^2}\)

Scenario return and risk

\[ \begin{align} E(R) &= \sum_{s=1}^4 p_s \times R_s \\ &= (0.25 \times 0.31) + (0.45 \times 0.14) + (0.25 \times -0.0675) + (0.05 \times -0.52) \\ &= 0.0976 \\ &=9.76\% \end{align} \]

Scenario return and risk

\[ \begin{align} \sigma^2 &= \sum_{s=1}^4 p_s \times [R_s-E(R)]^2 \\ &= 0.25 \times (0.31-0.0976)^2 + 0.45 \times (0.14-0.0976)^2 + \\ & \qquad 0.25 \times (-0.0675-0.0976)^2 + 0.05 \times (-0.52-0.0976)^2 \\ &= 0.038\\ \sigma &= \sqrt{0.038}=0.1949=19.49\%\\ \end{align} \]

Arithmetic and geometric averages

• Arithmetic average
  • If we assume the probability of each state is equal then \(p_s=\frac{1}{K}\): \[E(R) = \sum_{s=1}^K p_s \times R_s=\frac{1}{K}\sum_{s=1}^K R_s\]
• Geometric (time-weighted) average
  • Terminal value (TV) of the investment: \[TV_n = (1+r_1)(1+r_2)...(1+r_n)\]
  • Geometric average return \((g)\) over \(n\) time periods: \[g=TV_n^{1/n} -1\]

Estimating variance and standard deviation

• Population variance is the measured as the expected value of squared deviations \[\sigma^2 = \frac{1}{n} \sum_{t=1}^n [R_t-\bar{R}]^2\]
• We want to work with the sample variance and standard deviation so need to divide by \(n-1\) rather than \(n\) \[\sigma^2 = \frac{1}{n-1} \sum_{t=1}^n [R_t-\bar{R}]^2\] \[\sigma = \sqrt{\frac{1}{n-1} \sum_{t=1}^n [R_t-\bar{R}]^2}\]

The normal distribution

• Investment analysis can “appear” easier if we assume asset returns follow a normal distribution
  • Symmetric asset returns \(\rightarrow\) Standard deviation is a good measure of risk
  • Symmetric asset returns \(\rightarrow\) Portfolio returns will be symmetric as well
  • Only mean and standard deviation needed to estimate future scenarios
  • Pairwise correlation coefficients summarize the dependence of returns across securities
• You need to be aware that we are making an assumption when we focus only on means and standard deviation

The normal distribution

Investor decision making

• How does an investor choose the amount to invest in risky assets?
  • We can use a utility function to rank alternative outcomes
  • Investors choose among different assets or portfolios to maximise expected utility
  • Investors can have different levels of risk aversion
  • How much do they dislike risk?
  • The utility function will lead to different portfolio choices by investors
• If there is no uncertainty then we just need to determine how much we want to consume now, and how much we want to consume later
  • Risk-free asset
• If return is not certain across all possible states of the world
  • The range of possible future cash flows will impact on wealth
  • Risky assets

Portfolio with two risky assets

• For a portfolio with two risky assets, the expected return and variance equations are: \[E(R_p) = w_1 E(R_1) + w_2 E(R_2)\]

• Correlation coefficient between two assets:

• Need to estimate two expected return, two variances and one covariance/correlation

Optimal combined portfolio

• Choosing the optimal risky portfolio
  • Best combination of risky and safe assets to form portfolio
  • Optimal portfolio CAL tangent to efficient frontier
  • Slope of CAL is Sharpe Ratio of risky portfolio \[Sharpe\ Ratio = \frac{E(R_i)-R_f}{\sigma_i}\]
• Separation Property implies portfolio choice, separated into two tasks
  1. Determine the optimal risky portfolio that contains only risky assets
  2. Determine the allocation to risky portfolio and risk-free asset based on investor’s risk aversion (A)

Conclusion

• Return and risk refresher
• Use a utility function to model investor preferences
  • This allows us to combing risk and return into a single equation
  • Allows us to incorporate different levels of risk aversion (A)
• Portfolio theory
  • What are the risk and return of portfolios?
  • How do investors benefit from diversification?
  • What portfolios maximise investor utility?
    • Without a risk-free asset
    • With a risk-free asset

Portfolio risk and correlation